Built-in Mathematica Symbol

LegendreP

LegendreP[n, x]
gives the Legendre polynomial

.

LegendreP[n, m, x]
gives the associated Legendre polynomial

.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

Explicit formulas are given for integers and .

The Legendre polynomials satisfy the differential equation .

The Legendre polynomials are orthogonal with unit weight function.

The associated Legendre polynomials are defined by .

For arbitrary complex values of n, m and z, LegendreP[n, z] and LegendreP[n, m, z] give Legendre functions of the first kind.

LegendreP[n, m, a, z] gives Legendre functions of type a. The default is type 1.

The symbolic form of type 1 involves , of type 2 involves and of type 3 involves .

Type 1 is defined only for within the unit circle in the complex plane. Type 2 represents an analytic continuation of type 1 outside the unit circle.

Type 2 functions have branch cuts from to and from to in the complex plane.

Type 3 functions have a single branch cut from to .

LegendreP[n, m, a, z] is defined to be Hypergeometric2F1Regularized[-n, n+1, 1-m, (1-z)/2] multiplied by for type 2 and by for type 3.

For certain special arguments, LegendreP automatically evaluates to exact values.

LegendreP can be evaluated to arbitrary numerical precision.

LegendreP automatically threads over lists.

EXAMPLES

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Basic Examples (2)

Compute the 10

Legendre polynomial:

Compute the 10

Legendre polynomial:

Scope (7)

Compute the associated Legendre polynomial

:

Compute a half-integer associated Legendre function:

Evaluate for fractional orders:

Evaluate for complex orders and arguments:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

LegendreP threads element-wise over lists:

TraditionalForm formatting:

Generalizations & Extensions (4)

LegendreP can deal with real-valued intervals:

LegendreP can be applied to a power series:

Different LegendreP types give different symbolic forms:

Types 2 and 3 have different branch cut structures:

Applications (3)

Angular momentum eigenfunctions:

Find quantum eigenfunctions for modified Pöschl-Teller potential:

Generalized Fourier transform for functions on interval -1 to 1:

Properties & Relations (1)

Use FunctionExpand to expand into simpler functions:

Possible Issues (1)

Cancellations in the polynomial form may lead to inaccurate numerical results:

Evaluate the function directly:

Neat Examples (2)

Visualize distribution of zeros:

Generalized Lissajous figures: